Randomized experiments are widely used to estimate causal effects of proposed treatments in domains spanning the physical and biological sciences, social sciences, engineering, medicine and health, as well as in public service domains and the technology industry. However, classical approaches to experimental design rely on critical independence assumptions that are violated when the outcome of an individual may be affected by the treatment of another individual, referred to as network interference. This interference introduces computational and statistical challenges to causal inference. Our work focuses on estimating the Total Treatment Effect, informally described by the difference in average outcomes across the population when everyone versus no one is treated. We present a new hierarchy of low-degree polynomial potential outcomes models and unbiased estimators that enable statistically efficient and computationally simple solutions. When the network is completely unknown, we provide a simple estimator along with a staggered rollout randomized design which is unbiased and has low variance. To our knowledge, we are the first to propose a statistically grounded solution for the case where the underlying network is completely unknown. We evaluate the finite-sample performance of our proposed estimators relative to existing estimators on simulated data. Our proposed estimator has reduced mean squared error relative to previous approaches when the network is complex or unknown, or when the network interference effects are fully heterogeneous or nonlinear. Our approach also extends to other estimands and to settings beyond neighborhood interference.